You're about at the point (level 28) where Conjurer gets the trait to allow casting Raise while in-combat. From that point on, you're really going to want the ability to pair it with Swiftcast, so getting Thaumaturge up to 26 will be important. Somewhere between reaching 30 and doing the level 32 dungeon you're going to want to unlock your job, so you'll also need Arcanist up to at least 15. That would leave Eye for an Eye (at Arcanist level 34) as your only remaining cross-class to get. It's quite a bit more leveling just for one skill, and though quite a useful one, it's not as essential as Swiftcast is, so could be put off for a bit if you want to get back to focusing on your White Mage job directly. Perhaps aim for getting ACN the rest of the way to 34 by around the time your WHM hits 50. It's good to have your full suite of skills available for some of the level 50 dungeons.

If you're interested in getting into the crafting / gathering part of the game, it's a good idea to at least unlock those classes early on. One of the really good ways to get DoH/DoL experience (as well as a steady supply of GC seals) is by turning in crafted/gathered items to your Grand Company, but that opportunity is only offered for one item per day per class, so isn't something you can focus on and grind your way through. (There are of course grindable ways of leveling, but they tend to take more effort per XP.) The GC turn ins are better suited to spending a little time each day on that, while the bulk of your day is (usually, unless you're not on for long) spent on other things.

As to crafting being a money sink, it can be if you let it, but if you combine it with gathering classes as well, you can choose when ingredients are cheap enough to just buy and when they're expensive enough you'd rather just go get them yourself. It's either a money sink or a time sink (or most often a compromise with a little of each) but you get to choose whether money or time is more affordable at the moment. And if you spend a little extra time to gather or make stuff to sell, it can provide an income stream while you're leveling it.

As for whether to branch out further, picking up some DoW classes, it can give you some versatility, for instance if you have friends who want to run something but need a tank rather than a healer. It can give you some insight into how other classes and roles function, which might help you coordinate with your teammates more effectively. And it can give you some extra variety if you get bored with doing the same job all the time. But none of these are really direct benefits on how your White Mage is, if you're confident that's really going to be your primary job. It's probably more common for players to fully advance their primary job (along with enough of its supporting classes to get cross-class skills) all the way to endgame before branching out to unrelated jobs. The people who branch out earlier tend to either be still trying to decide which job they like the best, or they're simply more interested in the variety than they are in reaching endgame.